Effect of a vertical wall coupled with a rising bubble on the motion characteristics of a rising oil droplet in a two-dimensional Hele–Shaw cell

The coupled effects of a vertical wall and a rising bubble on the motion of an oil droplet are investigated in a two-dimensional Hele–Shaw cell with a 60 vol. % glycerol concentration. Centroid trajectories and dimensionless velocities show that the wall effect on the isolated oil droplet and bubble is localized, becoming negligible beyond the dimensionless wall distances So*=5.06 and Sb*=4.55. Under wall–bubble coupling, bubble-induced entrainment promotes oil droplet acceleration, whereas wall-induced resistance suppresses acceleration or enhances decay. Trajectory-inferred force balance shows that the drag-to-buoyancy ratio FD/FB remains near unity, whereas lateral displacement X and the lateral force-to-buoyancy ratio FL/FB vary strongly with So*. Comparing the oil–bubble and wall–oil–bubble cases shows that the vertical wall modulates, rather than amplifies, the bubble-induced lateral migration of the oil droplet.